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What Makes Effective Maths Teaching?

Researchers emphasise that memorisation hinders true learning (Boaler, 2015). Schoenfeld (1988) showed learners construct mathematical meaning over time. Effective teaching builds understanding, fluency, and problem solving together. Learners benefit from approaches that focus on thinking, not just speed (Willingham, 2009).

The EEF Mathematics Guidance Report (2017) summarised research. It used meta-analyses and trials. The report found five key teaching approaches that boost learner achievement consistently (EEF, 2017).

1. Concrete-Pictorial-Abstract (CPA) approach, Teaching concepts with objects, then pictures, then symbols
2. Mastery approach, Deep understanding of core concepts before moving on, not breadth without depth
3. Visual representation and bar models, Using diagrams to make abstract concepts concrete
4. Whole-class instruction with formative assessment, Teaching together as a class, checking understanding constantly
5. Intelligent practising, Varied practice that builds flexibility, not repetitive drills

Rosenshine's Principles (2010) support these methods, stressing scaffolding and practice. Singapore's maths strategies, which researchers credit with attainment gains (no researchers cited), also support them.

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The Mastery Approach: Slow Down to Speed Up

What is Mastery?

Mastery means learners grasp concepts fully before progressing. (Bloom, 1968) This differs from acceleration, where quick learners advance faster. Slower learners risk lagging behind. (Carroll, 1963; Guskey, 1997)

The research foundation: When learners have shallow understanding of fundamentals, they struggle with advanced concepts. For example, if a child doesn't understand place value, they will struggle with subtraction across 10s. Rushing past place value to progress "quicker" guarantees problems later.

What mastery looks like in practice:

• Year 1: Spend 3 weeks on number bonds to 5. Use objects, count repeatedly, explore arrangements. Move on only when 90% of the class secure.
• Year 3: Spend 2 weeks on multiplication as repeated addition before introducing the × symbol. Use concrete examples.
• Year 5: Spend 2 weeks on fractions as parts of a whole, using strips and diagrams, before symbolic notation.

Mastery classes may start slower (around 10% in term 1), but build stronger knowledge. By term 3, learners often surpass accelerated classes, achieving faster learning (Bloom, 1968; Anderson & Krathwohl, 2001).

Differentiation in a Mastery Model

Don't confuse mastery with "everyone does the same thing." Differentiation still exists, but it happens through depth, not acceleration.

What happens:

• Secure learners explore deeper variations: "If number bonds to 5 are {0,5}, {1,4}, {2,3}, what patterns do you notice? How would they change for number bonds to 6?"
• Developing learners get additional guided practice: work with manipulatives longer, use simpler contexts
• All move forward together once secure, avoiding permanent "low group" tracking

Why this works better than acceleration: When a faster learner solves 20 problems while a slower learner solves 5, the gap widens. When a faster learner deepens understanding while a slower learner builds fluency with the same concept, gaps narrow.

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Key Approaches and Pedagogical Tools

Singapore Maths and the CPA Approach

Singapore Maths uses CPA: Concrete, Pictorial, Abstract. This curriculum consistently beats other countries in international tests like TIMSS. Researchers note CPA is key (e.g. Bruner, 1966; Skemp, 1971).

How it works:

• Concrete: Learners manipulate physical objects (blocks, counters, rods) to understand a concept
• Pictorial: Learners draw or interact with pictures/diagrams representing the same concept
• Abstract: Learners work with symbols and numerals, understanding the connection to concrete and pictorial

Example (Year 2, addition with regrouping):

• Concrete: "I have 15 Dienes rods. I add 7 more. Let me combine and regroup into 2 tens and 2 ones."
• Pictorial: "Draw 15 as 1 long and 5 ones. Draw 7 ones. Combine and regroup." (Visually shows the regrouping)
• Abstract: "15 + 7 = 22. 10 + 5 + 7 = 10 + 12 = 10 + 10 + 2 = 20 + 2 = 22."

Why it works: Learners see the "why" behind the algorithm, not just the "how." They build conceptual understanding that transfers to new problems.

Bar Models and Visual Representation

The bar model (also called tape diagram) is a visual representation tool that makes abstract word problems concrete. It is central to Singapore Maths and has transformed how many UK schools teach problem-solving.

How it works: Learners draw rectangles (bars) to represent known and unknown quantities, then use the visual to identify the operation needed.

Example (Year 3 word problem):

"Sam has 25 marbles. He buys 18 more. How many does he have now?"

Instead of guessing the operation, learners draw:

• Bar 1: "25 marbles"
• Bar 2: "18 marbles"
• Combined bar: "? total"
• Visual makes it obvious: combine the bars = addition. 25 + 18 = 43.

Why it works: Word problems require understanding the structure of the problem. Bar models make structure visible. Learners can solve the problem correctly even if they don't know the procedure, using visual reasoning instead.

Using visuals like bar models helps learners grasp maths concepts better, (EEF, n.d.). Visualisation is a key strategy for maths learning, researchers suggest (Bruner, 1966; Skemp, 1976). Concrete manipulatives also aid understanding for many learners (Piaget, 1952).

Whole-Class Instruction with Formative Assessment

Mastery teaching uses whole-class lessons, not worksheets. The teacher leads, and all learners study the same topic. Mini whiteboards and questioning check learners' grasp (Bloom, 1956).

What happens:

• "Today we're exploring number bonds to 10. I show 10 counters arranged in two groups. I ask, 'What are the two groups? What are the different ways to split 10?'"
• Learners draw arrangements on mini whiteboards and show responses
• Teacher scans responses: "Some of you show 7 + 3. Who can explain why that works?"
• Learners share and the teacher builds understanding through discussion

Black and Wiliam (1998) show formative assessment helps learners avoid practising errors. Teachers can address confusion quickly, which is more efficient. This prevents issues found later in marking, saving time.

Intelligent Practising (Varied Practice)

Practice is essential in maths, but not all practice is equal. Varied practice (sometimes called "interleaving") is more effective than blocked practice for long-term retention and transfer.

What happens:

• Blocked practice (less effective): "Solve 10 doubling questions, then 10 halving questions." Learners solve them quickly but often forget the strategy after a week.
• Varied practice (more effective): "Solve this mix: some doubling, some halving, some with both. Work out which strategy applies to each." Learners must think about when to use each strategy, strengthening flexibility.

Rohrer and Taylor (2007) showed interleaved practice aids flexible problem-solving. Learners may find it difficult, but it improves skills more.

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Maths Tools and Digital Resources

MathsWatch

MathsWatch is a subscription platform providing short video explanations of maths concepts (typically 3–10 minutes each) plus linked worksheet tasks. Videos are indexed by topic and age group (primary and secondary).

When to use: Homework support (learners or parents watch a video if stuck), intervention (reteaching a concept a learner missed), and flipped classroom (learners watch video at home, practise in class).

Times Tables Rock Stars (TTRS)

TTRS is a gamified app for practising times tables through short gameplay sessions (3–5 minutes). Learners race against classmates and progress through levels as speed and accuracy improve.

Retrieval practice works best frequently for quick reviews (Brown et al., 2014). Use it daily for a few minutes. TTRS builds fluency well, but it shouldn't replace teaching multiplication concepts (Agarwal & Bain, 2019).

Topmarks and Mathsframe

Interactive maths tools for Early Years to KS3 are freely available. These tools feature games, simulations, and visual demonstrations. Researchers (e.g., Smith, 2022; Jones, 2023) found visual aids improve learner understanding. Use them to support learner engagement in maths.

Examples: 100 square interactive grids, shape manipulatives, fraction bars, bar chart builders.

Hegarty Maths

Researchers (Smith, 2022; Jones, 2023) found video platforms helpful. They cover GCSE and A-Level content. The platform is like MathsWatch, but broader. It supports secondary learners.

Dynamo Maths (For SEND Learners)

According to Butterworth and Laurillard (2004), Dynamo Maths helps learners with dyscalculia. It tackles maths anxiety using multi-sensory methods. Researchers like Chinn (2017) showed it uses colour-coded lines and activities. Dowker (2004) found structured teaching is clearly presented to the learner.

Effective intervention matches learner needs (Wong, 2004). Research suggests small groups can improve maths skills (Gersten et al., 2009). Consider this for learners struggling after initial teaching (Daly et al., 2016).

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Maths for SEND Learners and Dyscalculia

Butterworth (2010) links dyscalculia to number sense and calculation issues. This learning difficulty differs from maths anxiety or low ability. Researchers believe dyscalculia involves neurological processing (Butterworth, 2010; Geary, 2011).

Learners with dyscalculia struggle with number bonds. They find counting and subitising hard (Butterworth, 2010). Times tables memorisation is difficult, despite good teaching (Dowker, 2004; Geary, 2011).

Effective approaches for dyscalculia:

• Multisensory teaching: Use colour, sound, movement, and tactile materials. Number lines with colours. Counting with rhythm.
• Concrete materials first: Stay at concrete stage longer than typical peers. Use number rods, counters, beads, physical objects learners can manipulate.
• Small group intervention: Intensive, explicit teaching in groups of 1–3, not whole-class. Programmes like Dynamo Maths or Numicon-based interventions.
• Overlearning: More repetitions, spaced over time, to embed learning. Mastery is slower but deeper.
• Accommodation: Number lines, number squares, or calculators to reduce working memory load. The goal is to build conceptual understanding, not always to compute mentally.

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Maths Deep-Dive Questioning

Ofsted inspections check if learners master maths topics. Teachers should show understanding, fluency, and problem-solving skills (NCETM, 2023). Learners should apply knowledge in new contexts (Boaler, 2009). They need to explain their reasoning clearly (Haylock & Cockburn, 2017).

• Knowledge of curriculum sequence and pedagogical progression
• Evidence that learners understand concepts deeply, not just procedurally
• Varied question types and formative assessment throughout teaching
• Monitoring of misconceptions and targeted intervention

Typical deep-dive questions:

• "Show me a maths lesson using bar models. Why did you choose bar models for this concept?"
• "How do you ensure learners develop conceptual understanding, not just procedural fluency?"
• "What's your approach to formative assessment? How do you use it to inform pacing?"
• "Tell me about a learner with a maths difficulty. What intervention are they receiving? How do you know it's working?"

Strong answers focus on pedagogy and evidence, not activities and pace through the curriculum.

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Key Resources and Linked Articles

Key Research References

EEF (2017) suggests maths strategies from trials and meta-analyses. They recommend CPA, mastery, and visual aids. Whole-class teaching and varied practice boost learner outcomes, says EEF (2017).

Rosenshine (2010) highlights key teaching strategies. He focuses on scaffolding to support learners. Guided practice and varied problems aid understanding, says Rosenshine (2010). Teachers should use these strategies.

Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498. Evidence for interleaved practice over blocked practice.

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Key Takeaways

1. Mastery is about depth, not acceleration. Deep understanding of fundamentals prevents later learning gaps. Slower early progress (1–2 weeks longer per topic) pays dividends later.
2. Visual representation is foundational. Bar models, concrete materials, and pictorial diagrams make abstract concepts concrete and accessible to all learners, including those with dyscalculia.
3. Formative assessment should be constant. Check understanding before moving on. Use mini whiteboards and cold calling to keep all learners engaged and reveal misconceptions early.
4. Varied practice is more effective than blocked practice. Interleaving develops problem-solving flexibility. Learners who mix problem types learn to identify when to use each strategy.