See also: [[Reading Wars]] >[!example] Multiplication Facts >$3 \times \square = 50$ > >$7=63 \div \square$ > >A student who has [[Fluency|automaticity]] in their multiplication facts can near-instantly recognise the missing factor or divisor. Students in Year 7 should be fluent in their multiplication fact families up to $12 \times 12$, or at least $10 \times 10$ ## Cognitive benefits of automaticity Memorising multiplication facts reduces the [[Cognitive Load]] of complex mathematical problem-solving. When multiplication facts are stored in long-term memory, they require minimal cognitive resources to recall. This frees up working memory to focus on higher-order thinking and more complex tasks. ## The memorisation debate Some progressive education advocates argue against the memorisation of multiplication facts, suggesting that understanding the underlying concepts is more important than rote memorisation. They claim that students can derive multiplication facts from first principles or through the use of strategies such as skip counting. This represents a [[Logical Fallacies|false dichotomy]]. Conceptual understanding matters, but this stance overlooks the cognitive benefits of [[Fluency|automatic]] recall. Without automatic recall, students expend cognitive resources on basic calculations, leaving less capacity for more complex problem-solving and reasoning. Students who must derive 7 × 8 during algebraic equation solving cannot simultaneously attend to the algebraic structure. ## Strategies for teaching multiplication facts Instruction in multiplication facts requires sequencing and practice structures that build fluency. Teaching the commutative property halves the number of facts students must memorise. Including drills that pair facts, such as practising both 7 × 5 and 5 × 7 in the same session, emphasises this property and reduces the learning burden. Introduce facts gradually, starting with the easiest multiplication facts such as 0s, 1s, 2s, 5s, and 10s before moving to more challenging facts. Regular review of previously learnt facts reinforces retention. Scheduling review sessions for previously learnt multiplication facts, whether using spaced repetition software or simple periodic revisiting, maintains accessibility of facts in long-term memory. Teaching multiplication and division together as related operations helps students understand fact families. The fact family for 4 × 6 = 24 includes 6 × 4 = 24, 24 ÷ 6 = 4, and 24 ÷ 4 = 6. This helps students recognise multiplication and division as inverse operations. Incorporating division problems alongside multiplication problems in practice sessions, such as practising 4 × 6 and then 24 ÷ 6, reinforces these connections. ## Number bonds The ability to near-instantly recognise the missing addend or subtrahend forms the foundation for all arithmetic operations. Students in Year 7 should be fluent in their number bonds up to 100. For example, students should immediately recognise that 3 + ? = 10 requires 7, or that 35 - ? = 11 requires 24. ## References Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. *Psychological Bulletin*, 132(3), 354-380. https://doi.org/10.1037/0033-2909.132.3.354 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 Logan, G. D. (1988). Toward an instance theory of automatization. *Psychological Review*, 95(4), 492-527. https://doi.org/10.1037/0033-295X.95.4.492 National Mathematics Advisory Panel. (2008). *Foundations for success: The final report of the National Mathematics Advisory Panel*. U.S. Department of Education. 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 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 Woodward, J. (2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. *Learning Disability Quarterly*, 29(4), 269-289. https://doi.org/10.2307/30035554