## Key Ideas > [!abstract] Core Concepts > > - Students need fluent declarative and procedural knowledge before tackling unfamiliar problems > - Explicit instruction is efficient; means-end analysis is cognitively demanding > - Domain-specific knowledge, not generic skills, determines problem-solving success ## Definition **Problem-solving**: 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]] --- ## Problem definition 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. ## Knowledge acquisition methods Students can acquire knowledge through two methods with different cognitive demands. **Explicit instruction** involves learning from others through direct teaching and 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 process 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 requires cognitive resources due to high [[Element Interactivity|element interactivity]], as 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|surface and deep structure]] in problems, leading to [[Cognitive Load|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: when prior experience hinders problem-solving 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. This finding highlights risks of over-practising specific procedures without developing conceptual understanding. 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 and knowledge Experts solve problems more effectively than novices (Chi, Feltovich, & Glaser, 1981). Experts 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). ## Generic problem-solving heuristics Common strategies taught in schools include 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). > [!tip] Implications for Teaching > > - If you want your 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 students [[Explicit Teaching|explicitly]] before getting them to solve 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