## Key Ideas > [!abstract] Core Concepts > > - Students often need fluent procedures before they can engage with deeper conceptual explanations > - Advanced mathematical reasoning can exceed working memory capacity for learners without sufficient foundation > - GeoGebra and other visual demonstrations make concepts accessible when formal proofs are inappropriate ## Definition **Procedural Alongside Conceptual**: Teaching approach that develops procedural fluency and conceptual understanding in parallel, often beginning with procedures to enable later conceptual engagement. ## Connected To [[Explicit Teaching]] | [[Cognitive Load Theory]] | [[Fluency]] | [[Worked Examples]] | [[Part-whole approach]] | [[Concrete Pictorial Abstract]] --- ## The challenge of conceptual explanations Teaching advanced mathematical reasoning to novices can exceed working memory capacity when prerequisite knowledge is insufficient. Teaching volumes of revolution to Year 9 students creates cognitive overload because students lack the necessary mathematical foundation (Sweller et al., 2019). Conceptual explanations that require multiple sophisticated ideas simultaneously exceed novice processing capacity (Cowan, 2001). Understanding why something works requires more cognitive resources than learning how to do it. This challenge appears across mathematics topics. Proving sphere volume using calculus concepts, deriving the quadratic formula before students have algebraic fluency, or explaining statistical distributions without computational confidence all create difficulties. In each case, the conceptual explanation demands cognitive resources that learners have not yet developed. ## Visual demonstrations and procedural foundations When formal proofs exceed student capacity, visual demonstrations offer an alternative route. Interactive visualisations through tools like GeoGebra allow students to explore mathematical relationships without formal derivation requirements (Ainsworth, 2006). Physical manipulatives and visual representations make abstract concepts accessible. These approaches focus on why mathematical ideas make sense rather than providing rigorous mathematical justification. Alongside visual tools, procedural fluency creates a foundation for later conceptual engagement. Students develop computational confidence before engaging with conceptual complexity. Beginning with procedures - how to do something - precedes explanations of why it works. As procedural knowledge becomes automated, teachers can increase conceptual depth without overwhelming working memory. ## Sequencing procedural and conceptual learning Teachers must assess student readiness before introducing conceptual explanations. Teachers must ensure students have necessary foundational knowledge and monitor cognitive load by watching for confusion, frustration, or disengagement. Individual differences matter: some students are ready for conceptual work whilst others need more procedural practice. The typical sequence begins with developing procedural fluency through guided practice and repetition. Teachers then introduce intuitive explanations using visual and concrete supports. As the foundation strengthens, they gradually increase conceptual sophistication. Finally, students integrate their understanding through problem-solving and applications. Supporting tools enhance this progression. Visual software can demonstrate concepts beyond students' current mathematical level. Multiple representations show the same idea through different modalities: numerical, visual, and algebraic. The concrete-pictorial-abstract progression moves systematically from hands-on to symbolic understanding (Fyfe et al., 2014). ## The relationship between procedures and concepts The procedural-first approach reduces working memory demands during conceptual learning. It provides a foundation of success before challenging material and enables students to focus on reasoning rather than computational struggle. This approach builds confidence through procedural competence and provides tangible progress markers. It prevents cognitive overload during concept introduction, allows deeper engagement with mathematical ideas, and supports transfer to new problem contexts. Teaching procedures first does not mean avoiding conceptual development. It means sequencing appropriately for cognitive load management. Procedural practice should be purposeful and connected to larger mathematical ideas, not mindless repetition. Both procedural and conceptual knowledge are essential and mutually reinforcing (Rittle-Johnson et al., 2001). Students need both fluent procedures and conceptual understanding for mathematical competence (Star, 2005). Procedural fluency supports conceptual learning, whilst conceptual understanding enriches procedural application (Rittle-Johnson et al., 2001). The optimal sequence depends on context: some concepts benefit from a procedures-first approach, others from concepts-first, depending on complexity and prerequisites. ## 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 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 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 Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. *Journal of Educational Psychology*, 93(2), 346-362. https://doi.org/10.1037/0022-0663.93.2.346 Star, J. R. (2005). Reconceptualizing procedural knowledge. *Journal for Research in Mathematics Education*, 36(5), 404-411. Sweller, J., van Merriënboer, J. J. G., & Paas, F. (2019). Cognitive architecture and instructional design: 20 years later. *Educational Psychology Review*, 31(2), 261-292. https://doi.org/10.1007/s10648-019-09465-5