Key ideas
Core concepts
- Students need fluent declarative and procedural knowledge before tackling unfamiliar problems, so teach the prerequisites explicitly first.
- Explicit instruction places low demands on cognitive resources; means-end analysis is cognitively demanding and suited to applying knowledge, not learning it. Domain-specific knowledge, not generic skills, determines problem-solving success.
Problem-solving is the ability to solve unfamiliar problems by leveraging connections between concepts within existing knowledge structures (schemas) (Chi, Feltovich, & Glaser, 1981).
Connected to
Element Interactivity | Surface and Deep Structure | Biologically Primary & Secondary Knowledge | Cognitive Load Theory | Explicit Teaching | Schema | Experts and Novices Think Differently | 21st Century Skills
According to Cognitive Load Theory, a problem is any situation where a goal state must be reached from a given initial state by performing a series of mental operations (Sweller, 1988). Problems can be familiar or unfamiliar, routine or non-routine.
Students can acquire knowledge through two methods with different cognitive demands. Explicit instruction, learning from others through direct teaching, places low demands on cognitive resources. Means-end analysis requires making educated guesses based on prior experience and imposes high cognitive demands, making it suitable only for applying established knowledge rather than learning new content.
Means-end analysis
Means-end analysis, the natural problem-solving strategy people use when lacking explicit knowledge, involves generating and testing problem-solving steps against reality (Sweller, Mawer, & Ward, 1983). This consumes cognitive resources because of high element interactivity: learners must simultaneously hold in working memory the current state of the problem, the goal state they’re trying to reach, the relationship between current and goal states, and potential moves with their consequences (Cowan, 2001). Novices struggle to differentiate between surface and deep structure in problems, leading to cognitive overload (Chi, Feltovich, & Glaser, 1981). Their working memory fills with means-end processing, leaving insufficient capacity for learning (Sweller, 1988).
Students succeed when they can solve unfamiliar problems by searching for solutions selectively rather than randomly, but this selectivity requires domain knowledge (Chi, Glaser, & Rees, 1982).
The Einstellung effect
Established patterns of thinking can interfere with finding simpler or more efficient solutions. Luchins (1942) demonstrated this in the water jug experiment, where participants measured specific quantities of water using three jugs of different capacities. After solving several problems using a complex method (B-2C-A), participants persisted with this approach even when simpler solutions became available. Prior experience with a particular solution method created mental rigidity, preventing recognition of more efficient alternatives.
Over-practising specific procedures without developing conceptual understanding therefore carries risks: students may apply familiar methods mechanically without considering whether simpler approaches exist. Instruction should include varied problem types and encourage flexibility in problem-solving rather than promoting rote application of single methods. Teachers should help students understand when and why particular methods apply, supporting both proceduralisation and conceptual understanding (Luchins, 1942).
Expertise
Experts solve problems more effectively than novices (Chi, Feltovich, & Glaser, 1981). They possess domain-specific knowledge (Chi, Glaser, & Rees, 1982), develop more accurate and efficient solution strategies (Ericsson & Kintsch, 1995), recognise problem patterns quickly (Chase & Simon, 1973), and organise schemas more effectively for retrieval (Chi, Feltovich, & Glaser, 1981). This expertise results from extensive deliberate practice, not inherent talent (Ericsson, Krampe, & Tesch-Römer, 1993).
Schools commonly teach generic problem-solving heuristics: drawing a diagram, making a table, solving a simpler problem, looking for a pattern, working backwards, and guessing and checking. These heuristics are less effective without strong domain knowledge to guide their application (Sweller, 1988).
The teaching consequence follows directly. If you want students to learn to solve problems, they first need both the declarative and procedural knowledge within the subject area of the problem in question (Chi, Feltovich, & Glaser, 1981). Ensure students are fluent in all prerequisite skills by teaching them explicitly before setting unfamiliar problems (Rosenshine, 2012).
References
Chase, W. G., & Simon, H. A. (1973). Perception in chess. Cognitive Psychology, 4(1), 55–81. https://doi.org/10.1016/0010-0285(73)90004-2
Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5(2), 121–152. https://doi.org/10.1207/s15516709cog0502_2
Chi, M. T. H., Glaser, R., & Rees, E. (1982). Expertise in problem solving. In R. J. Sternberg (Ed.), Advances in the psychology of human intelligence (Vol. 1, pp. 7–75). Erlbaum.
Cowan, N. (2001). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24(1), 87–114. https://doi.org/10.1017/S0140525X01003922
Ericsson, K. A., & Kintsch, W. (1995). Long-term working memory. Psychological Review, 102(2), 211–245. https://doi.org/10.1037/0033-295X.102.2.211
Ericsson, K. A., Krampe, R. T., & Tesch-Römer, C. (1993). The role of deliberate practice in the acquisition of expert performance. Psychological Review, 100(3), 363–406. https://doi.org/10.1037/0033-295X.100.3.363
Luchins, A. S. (1942). Mechanization in problem solving: The effect of Einstellung. Psychological Monographs, 54(6), i-95. https://doi.org/10.1037/h0093502
Rosenshine, B. (2012). Principles of instruction: Research-based strategies that all teachers should know. American Educator, 36(1), 12–19.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285. https://doi.org/10.1207/s15516709cog1202_4
Sweller, J., Mawer, R. F., & Ward, M. R. (1983). Development of expertise in mathematical problem solving. Journal of Experimental Psychology: General, 112(4), 639–661. https://doi.org/10.1037/0096-3445.112.4.639