## Key Ideas > [!abstract] Core Concepts > > - **Multiple Representations**: Mathematical concepts understood through concrete, pictorial, and abstract forms with each offering unique insights > - **Progressive Abstraction**: Students should move toward abstract models as concrete or pictorial have limitations and time constraints > - **Representation Bias**: Teachers favour symbolic whilst students prefer pictorial; effective teaching requires fluent movement between all forms ## Definition **Concrete Pictorial Abstract (CPA)**: Teaching approach using multiple mathematical representations (physical materials, visual diagrams, and symbols) to build conceptual understanding. ## Connected To [[Solution Comparison]] | [[Teach Methods that Last]] | [[Knowledge-Based Curriculum]] | [[Prior Knowledge]] --- ![[ConcretePictorialAbstract.png|600]] ## Theoretical foundation Mathematical ideas, principles, and relationships can be expressed verbally, visually, and symbolically (Bruner, 1966). Each representation offers unique insights into concepts (Ainsworth, 2006). Effective mathematics teaching requires students to move fluently between multiple representations of the same idea (Pape & Tchoshanov, 2001). Bruner (1960) proposed that learning should progress through three modes of representation: enactive (learning by doing through physical manipulation), iconic (learning through images and diagrams), and symbolic (learning through abstract symbols and language). The concrete-pictorial-abstract approach aligns with this progression, recognising that students construct understanding more readily when beginning with physical experience before moving to visual representations and finally abstract symbols. The spiral curriculum principle suggests revisiting important concepts at increasing levels of sophistication, moving from concrete to abstract over time. ## Understanding new concepts requires concrete knowledge People understand new things in the context of things they already know, and most of what people know is concrete (Willingham, 2009). Abstract concepts prove difficult to understand without concrete foundations. Students cannot understand metaphors or analogies when they lack understanding of the concrete elements being referenced. "Mitochondria are the powerhouse of the cell" means nothing to students who do not understand what powerhouses do or how power generation works. New abstract ideas connect to existing concrete knowledge through analogies, examples, and models. However, these connections work only when students possess the necessary concrete understanding. Teachers must ensure students have the concrete knowledge required before introducing abstract concepts. This does not mean avoiding abstraction, but rather building it systematically from concrete foundations that students actually possess. The progression from concrete to abstract should be deliberate and evidence-based. Teachers should assess whether students have the concrete knowledge needed to understand planned analogies or examples, introduce concrete representations first before moving to abstract symbols, check that students understand the concrete elements before expecting them to grasp abstract concepts, and provide multiple concrete examples rather than relying on a single analogy that may not connect with all students. ## Representation preferences and biases Student and teacher preferences create predictable patterns that can either support or hinder learning. Recognising these biases enables deliberate teaching decisions that counteract limitations. Low-achieving students tend to prefer pictorial representations whilst high-achieving students often favour symbolic representations (Koedinger et al., 2008). Students generally gravitate toward familiar representation types. Mathematics teachers show bias toward symbolic representations (Nathan, 2012). Symbolic forms are viewed as more "mathematical" and worthy of classroom time. Teachers encourage symbolic work even when not most efficient, undermining strategic representation selection (Nathan, 2012). Teacher preferences shape student perceptions of mathematical validity, creating self-reinforcing cycles where symbolic approaches dominate regardless of appropriateness. ## Limitations of concrete and pictorial models Concrete and pictorial representations provide conceptual foundations but contain inherent limitations (McNeil & Jarvin, 2007; Uttal et al., 1997). Time constraints increase as quantities become larger, making models inefficient for complex calculations. Representation gaps occur when certain concepts cannot be modelled adequately (negatives are poorly shown in bar models). Overgeneralisation happens when fraction circles always show equal divisions, leaving students struggling with real-world unequal divisions. Complexity boundaries exist where models break down with advanced concepts, limiting transferability to higher mathematics. Students require extensive practice in fluently moving between representations, but should ultimately progress toward more abstract models (Fyfe et al., 2014; Rittle-Johnson & Alibali, 1999). Symbolic representations handle larger quantities and complex operations more efficiently. Abstract forms apply across broader mathematical contexts. Higher-level concepts require symbolic manipulation (Nathan, 2012). Expert mathematicians work primarily with abstract representations. ## Strategic model selection Selecting the appropriate model requires understanding the mathematical concept being taught and the affordances of each representation type. Number lines work best for rational number operations, fraction concepts, and integer arithmetic (Fuson & Briars, 1990). They are versatile tools for illustrating operations with rational numbers, help students recognise fractions as rational numbers rather than discrete quantities, and show relationships between different number types. Applications include addition or subtraction of integers, fraction ordering, and decimal placement. Integer counter models work best for integer operations and zero-pair concepts. Two different coloured counters represent positive and negative integers, illustrating zero-pairs concept clearly and supporting addition or subtraction of integers. However, the model becomes unwieldy with larger numbers. Bar models work best for linear equations, fact families, and proportional reasoning. They illustrate relationships in word problems effectively, bridge arithmetic and algebraic thinking, and provide clear representation of known and unknown quantities. Applications include simple linear equations like 3x + 5 = 17, fact family relationships, and part-whole problems. The critical limitation: bar models become complex with negative numbers. Transition to symbolic before introducing negatives. ![[BarModelEquations.png|400]] Area models work best for distributive law, polynomial operations, and factorisation. Applications include binomial expansion like (x + 3)(x + 5), trinomial expansions, factorisation, and even surds and imaginary numbers. The model must be configured specifically for each type of problem. Example progression: simple multiplication like 23 × 15, binomial products like (x + 4)(x + 7), then factorisation like x² + 11x + 28. ![[AreaModelExpanding1.png|400]] ![[AreaModelExpanding2.png|400]] ![[AreaModelFactorising.png|400]] Ratio boxes work best for proportional reasoning and similar figures. Applications include setting up proportional equations, solving ratio problems, similar triangle calculations, and scale factor problems. The structure provides clear visual representation of relationships between quantities. ## Implementation strategy Plan representation sequence by assessing prior knowledge of different representation types, selecting appropriate starting point based on concept complexity and student familiarity, planning progression pathway from concrete through pictorial to abstract, identifying transition points where students are ready for next representation level, and building fluency in moving between representations before advancing. Introduce representations systematically. Do not assume students automatically see connections between representations. Show how the same concept appears in each form, practice translating between representations, and discuss advantages and limitations of each approach. Use [[Solution Comparison]] to examine different approaches to the same problem. ## Common implementation errors Avoid over-reliance on concrete representations by planning clear progression toward abstraction based on student readiness rather than keeping students with manipulatives too long. Avoid representation isolation by explicitly demonstrating relationships between concrete, pictorial, and abstract forms rather than teaching each representation separately. Avoid premature abstraction by ensuring solid foundation in concrete or pictorial before advancing to symbols. Avoid single method teaching by deliberately incorporating all three forms and respecting different student learning preferences rather than only using teacher's preferred representation type. ## References Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. *Learning and Instruction*, 16(3), 183-198. https://doi.org/10.1016/j.learninstruc.2006.03.001 Bruner, J. S. (1960). *The process of education*. Harvard University Press. Bruner, J. S. (1966). *Toward a theory of instruction*. Harvard University Press. Fuson, K. C., & Briars, D. J. (1990). Using a base-ten blocks learning and teaching approach for first- and second-grade place-value and multidigit addition and subtraction. *Journal for Research in Mathematics Education*, 21(3), 180-206. https://doi.org/10.2307/749373 Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. *Educational Psychology Review*, 26(1), 9-25. https://doi.org/10.1007/s10648-014-9249-3 Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving. *Cognitive Science*, 32(2), 366-397. https://doi.org/10.1080/03640210701863933 McNeil, N. M., & Jarvin, L. (2007). When theories don't add up: Disentangling the manipulatives debate. *Theory Into Practice*, 46(4), 309-316. https://doi.org/10.1080/00405840701593899 Nathan, M. J. (2012). Rethinking formalisms in formal education. *Educational Psychologist*, 47(2), 125-148. https://doi.org/10.1080/00461520.2012.667063 Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding. *Theory Into Practice*, 40(2), 118-127. https://doi.org/10.1207/s15430421tip4002_6 Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? *Journal of Educational Psychology*, 91(1), 175-189. https://doi.org/10.1037/0022-0663.91.1.175 Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. *Journal of Applied Developmental Psychology*, 18(1), 37-54. https://doi.org/10.1016/S0193-3973(97)90013-7 Willingham, D. T. (2009). *Why don't students like school? A cognitive scientist answers questions about how the mind works and what it means for the classroom*. Jossey-Bass. ---