# Teach Methods That Last ## Key Ideas > [!abstract] Core Concepts > > - **Forward-facing vs Backward-facing**: Teach methods that transfer to complex questions rather than work only for simple endpoints > - **Unlearning is harder**: Reforming long-term memory connections is more onerous than learning correctly first time > - **Avoid tricks**: Teaching shortcuts that break down later and require complex rules for advanced problems ## Definition **Teach Methods That Last**: Teaching approaches that transfer across to complex questions rather than rules and tricks that fall apart later. ## Connected To [[Concrete Pictorial Abstract]] | [[Knowledge-Based Curriculum]] | [[Teaching is Not a Profession]] --- ## Forward-facing vs backward-facing methods The distinction between methods that serve immediate goals versus those that build lasting understanding can be seen in England's National Numeracy Strategy. In 1995, England launched a substantial initiative to improve primary mathematics, which later became the National Numeracy Strategy. The programme produced an initially sharp increase in attainment, but these improvements failed to continue through secondary examinations (Brown et al., 2003; Tymms, 2004). The problem lay in primary teacher training that was 'backward-facing' - methods were suitable for reaching a known endpoint but didn't progress students towards advanced mathematical ideas like algebra (Askew et al., 1997). An effective mathematics education system focuses on teaching 'forward-facing' approaches that maximise subsequent progression rather than pursuing short-term success (Gelman & Williams, 1998). ## Problematic methods > [!Example] Chunking Method > Repeatedly subtracting multiples of divisor from dividend until remainder less than divisor. > > 196 ÷ 6: Subtract 60 three times (leaving 16), then subtract 12, giving 32 remainder 4. > > Becomes cumbersome with larger numbers and doesn't generalise to algebraic division. > [!info] Backtracking "I'm Thinking of a Number" > Popular method: "I multiply by 2, subtract 1, get 11. What's my number?" > > ![[Backtracking.png|300]] > > Works for complex equations like $\frac{4(2x-3)}{5}+6$ but stops working with variables on both sides. > > When students encounter $3x-2 = x+4$, they search for relevant schemas but find none matching. Results in trial-and-error or misconceptions. ![[Equations Flowchart.png]] _Author's reflection: "Complex flowchart from my first year. Today, I'd spend weeks building proper foundations from Year 7."_ ## Tricks vs memory aids Not all mnemonics and shortcuts are equally problematic. The key distinction lies in whether they support or replace conceptual understanding (Rittle-Johnson et al., 2001). SOH CAH TOA for trigonometry aids recall without undermining understanding of the underlying relationships. In contrast, FOIL for binomial multiplication bypasses foundational understanding of the distributive property, leading to misconceptions when students encounter more complex expressions (Matz, 1982). ## Implementation principles Teaching methods that last require a coherent journey where each stage prepares students for the next level. Primary teachers bear responsibility for ensuring pupils are ready for secondary mathematics, while secondary teachers prepare students for higher-level mathematics. Professional standards require using proven methods rather than personal inventions. ## References Askew, M., Brown, M., Rhodes, V., Wiliam, D., & Johnson, D. (1997). *Effective teachers of numeracy: Report of a study carried out for the Teacher Training Agency*. King's College London. Brown, M., Askew, M., Millett, A., & Rhodes, V. (2003). The key role of educational research in the development and evaluation of the National Numeracy Strategy. *British Educational Research Journal*, 29(5), 655-672. https://doi.org/10.1080/0141192032000133679 Gelman, R., & Williams, E. M. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In D. Kuhn & R. S. Siegler (Eds.), *Handbook of child psychology: Vol. 2. Cognition, perception, and language* (5th ed., pp. 575-630). Wiley. Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman & J. S. Brown (Eds.), *Intelligent tutoring systems* (pp. 25-50). Academic Press. 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 Tymms, P. (2004). Are standards rising in English primary schools? *British Educational Research Journal*, 30(4), 477-494. https://doi.org/10.1080/01411920410001734264