My new article on reading fluency. Dyad reading, paired fluency and problems with silent reading.

Well-designed variation is not a trick or gimmick. It is evidence-informed teaching that builds mathematical insight by shaping what students see, not just what they do. Follow me if you enjoyed the thread, and subscribe to my free newsletter. Link in bio ✅. 10/10.

In practice, this means fewer worksheets packed with mixed questions, and more purposeful sequences that draw attention to structure, contrast, and change. The goal is not repetition, but perceptual clarity. 9/10.

Empirical support includes Heinze et al. (2009), who found that example-based learning with structural variation has a moderate positive effect on conceptual understanding in mathematics. Effectiveness increased when variation was planned, not incidental. 8/10.

This approach aligns with perceptual learning theory: novices do not automatically attend to the relevant variables in a task. Variation makes those variables impossible to ignore. (Marton & Booth, 1997; Gibson, 1969). 7/10.

In the commutativity example, we see separation (only order changes) and fusion (once several examples are seen together). This is how learners abstract the general property: a + b = b + a. 6/10.

Marton & Tsui (2004) describe four main types of variation:.•Contrast: show what something is and isn’t.•Separation: vary one thing, hold others constant.•Generalisation: expose common structure across diverse surface forms.•Fusion: multiple variations together. 5/10.

This isn’t just about noticing a pattern, it’s about internalising a mathematical property through well-structured experience. Variation guides attention to the invariant (the total) and the variant (the addends’ order). 4/10.

Consider this example adapted from Watson & Mason (2006):. To teach that addition is commutative:. Present:.3 + 7 = 10.7 + 3 = 10.2 + 9 = 11.9 + 2 = 11. The structure stays constant. The order changes. Students discern that the sum remains the same. 3/10.

Variation theory proposes that understanding improves when we systematically vary examples to highlight key features of a concept. Learning occurs not from exposure alone, but from discernment of critical aspects. 2/10.

Many students struggle in maths not because they can’t calculate, but because they don’t see what matters. Variation theory offers a research-informed method to guide attention where it matters. A thread inspired from a @BrunoReddyMaths & @rastokke podcast 🧵⬇️.

Must follow accounts for anyone interested in the science of learning:. @rastokke .@MrZachG .@BrunoReddyMaths .@Brad_Nguyen_ .@DTWillingham.@learnwithmrlee .@jettybe3 .@amandavande1 .@C_Hendrick .@jdtdobson .@BenisonMrs .@DaveMorkunas .@greg_ashman .@PepsMccrea .@tombennett71.

Controversial teacher opinion: . Every primary student should have a maths textbook. It should be low variance, with built in review, worked examples and faded guidance. Comment why I’m wrong or share if you agree.

Unfortunately, behaviour is the main thing stopping a lot of great teaching practices being implemented. Would like to see the SoL crowd discussing this more.

In maths, fluency of procedures isn’t the enemy of reasoning. It’s the foundation. I’m excited to listen to the Matific debate between Toni Hatten-Roberts & Pamela Weber Harris.

If you enjoyed this thread, retweet to share with others. Follow for more threads unpacking each of the 10 Rosenshine Principles of Instruction. 12/12.

Rosenshine’s principle reflects a deep truth from cognitive science. Learning is not exposure. It is repeated, effortful retrieval over time. Daily review builds the foundation on which durable knowledge, confidence and reasoning rest. 11/12.

Daily review isn’t optional for novice learners. When students lack fluency, their working memory is overloaded. They struggle to apply ideas, solve problems or think critically. Daily review helps build the fluency needed for higher-order thinking. 10/12.

Importantly, review should be cumulative. Not just yesterday’s material, but regular returns to prior knowledge from weeks or months ago. This creates interleaving, where knowledge from different times and topics connects into a coherent web. 9/12.

What might daily review look like?. • a slide deck relating to recent learning (use success criteria from previous lessons).• quick quiz answered on MWB’s, choral response or cold call.• revisiting a common misconception.• explaining a concept to a peer and share. 8/12.

Daily review also serves as a diagnostic tool for the teacher. It exposes gaps, confusions or forgotten content before they compound. Without this feedback loop, lessons float forward, while cracks in understanding silently widen. 7/12.