## Key Ideas > [!abstract] Core Concepts > > - **Surface features distract from principles**: Specific contexts, diagrams, and details can mislead students from underlying mathematical concepts > - **Deep structure reveals solution method**: Fundamental principles and operations required transcend surface variations > - **Expertise enables discrimination**: Experts recognise patterns while novices get caught up in irrelevant contextual details ## Definition **Surface and Deep Structure**: Surface structure refers to contextual details and specific scenarios in problems, while deep structure refers to the underlying mathematical principles required for solution. ## Connected To [[Experts and Novices Think Differently]] | [[Problem-Solving]] | [[Minimally Different Questions]] | [[Schema]] | [[Worked Examples]] | [[Interleaving Effect]] --- ## Definitions and examples The distinction between surface and deep structure affects teaching and learning. Surface structure refers to particular examples and scenarios designed to illustrate a mathematical principle (Chi, Feltovich, & Glaser, 1981). It includes specific details and contexts that vary from one problem to another: the story, the numbers, the diagrams, and the wording. Deep structure refers to the underlying mathematical principle that transcends specific examples and scenarios. It is the fundamental concept or rule that applies across different contexts - the actual mathematics required (Chi, Feltovich, & Glaser, 1981). > [!example] Example Problem Analysis ![[SurfaceDeepStructure.png]] > > In a typical word problem, surface structure includes details like 'age', 'school', and 'average' - specific to the problem context and potentially distracting novices from the actual task. Deep structure consists of the mathematical principle of finding $\frac{11}{9}$ of 36, which requires discerning relevant operations from surface details. Novices focus on surface features while experts identify the underlying structure. ## Expert vs novice differences [[Experts and Novices Think Differently|Experts]] can discern deep structure from surface structure, which is a key [[Problem-Solving|problem-solving]] skill (Chi, Glaser, & Rees, 1982; Sweller, 1988). When categorising problems, novices rely on surface features such as contexts and diagrams, while experts categorise based on deep structure and underlying principles. This difference emerges from novices' limited experience with pattern recognition compared to experts' extensive practice with related questions. Novices also tend to store isolated facts and procedures in memory, whereas experts possess organised knowledge that enables quick deep structure identification (Chi, Glaser, & Rees, 1982). Three factors account for expert advantages in discerning deep structure. First, extensive practice with related questions helps experts recognise patterns and principles across varied contexts (Ericsson, Krampe, & Tesch-Römer, 1993). Second, a significant volume of knowledge stored in long-term memory supports their understanding of fundamental concepts (Chase & Simon, 1973). Third, experts have their knowledge organised in a way that allows them to quickly identify the deep structure of problems, avoiding distractions from surface features (Chi, Glaser, & Rees, 1982). ## Research evidence > [!info] **Chi et al. (1981) Study** Chi investigated how experts and novices categorise physics problems (Chi, Feltovich, & Glaser, 1981). The experts were PhD students, while the novices were undergraduates. Both groups were given a set of physics problems to categorise based on their own criteria. Novices categorised problems based on surface features, such as whether a slope was involved. Experts categorised problems based on the physical principles required to solve them, such as the law of conservation of energy. Additional research confirms these findings across multiple domains (Chi, Glaser, & Rees, 1982; Sweller, 1988). Novices focus on superficial problem features while experts identify underlying solution structures (Sweller, Mawer, & Ward, 1983). ## Teaching implications Teachers can support students in developing the ability to identify deep structure through several approaches. Providing a variety of examples that illustrate the same deep structure helps students learn to generalise principles across different contexts (Paas & van Merriënboer, 1994). When teaching new procedures, problems that minimise irrelevant contextual details reduce surface distractions. Using [[Minimally Different Questions]] helps students focus on structural similarities. Gradually increasing the complexity of problems as students become more skilled at identifying deep structures supports learning (Rosenshine, 2012). Start with simple examples and progressively introduce more challenging ones. After solving problems, asking students to reflect on the deep structure they used and how it applies to other problems reinforces this skill (Chi et al., 1989). Having students practise categorising problems based on deep structure develops the skill of identifying relevant principles (Chi, Feltovich, & Glaser, 1981). Directly teaching students to identify underlying principles before attempting solutions provides necessary scaffolding. Building organised knowledge networks supports rapid pattern recognition. However, several limitations affect implementation. Students without sufficient domain knowledge cannot reliably identify deep structure (Chi, Feltovich, & Glaser, 1981). Surface similarities can trigger inappropriate solution attempts (Sweller, 1988). Context-heavy problems may overwhelm novice learners' working memory (Cowan, 2001). Pattern recognition requires extensive practice with varied examples (Ericsson, Krampe, & Tesch-Römer, 1993). These principles apply across mathematical topics. In quadratic word problems, surface features include contexts such as projectiles, revenue, or area, but deep structure is always a quadratic relationship requiring factorisation or the quadratic formula. Fraction problems display varying surface features like pizza slices, measurements, or ratios, but deep structure involves part-whole relationships and equivalent fractions. Linear equation contexts present surface features including money, time, or distance scenarios, but deep structure requires understanding variables, constants, and inverse operations. --- ## 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. 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